This technical note develops energy-event-triggered hybrid supervisory control techniques to address robust and fast energy equipartition for cyber-physical network systems, and discusses the application of the proposed approach to power systems. First, we present a hybrid controller with a distributed feedback and supervisory energy-event-triggered resetting law to achieve the robust disturbance rejection performance of physical networks by mimicking thermodynamic systems. The proposed controller architectures are constructed in such a way that each controller has a one-directional energy transfer from a plant to itself, and exchanges energy with its neighboring controllers. Specifically, if the cyber-physical system is lossless, this controller can prevent cascading behaviors. Second, we propose a consensus hybrid controller that mitigates the disturbance effect by decentralizing the disturbance on one plant to all the other plants when stabilizing it. In addition, a new combined hybrid controller based on the two precedent hybrid controllers is proposed to achieve cascade prevention and fast energy equipartition for lossless cyber-physical network systems. Finally, we apply our hybrid control technique to power systems, and simulation studies are carried out to show the efficacy of the proposed approach.